Two operators with the same button, it's possible?
Check Text Editor Templates: Operator Simple and Operator Modal

Imagine the modal operator has bl_idname = "object.some_generic_modal". In the execute function of your other (non modal) operator you could make a direct operator call:

def execute(self, context):
    # your stuff

    # call the other operator
    bpy.ops.object.some_generic_modal()

    return {'FINISHED'}


However, to avoid the separation and extra boilerplate, you might want to stick your one-off code in the invoke of the modal operator and call that directly instead.

To clarify: with the term one-off code I meant the code which just needs to be executed once, prior to the continuous modal operator.